A Mathematical Analysis of Discrete Waveform Relaxation Algorithms for Transmission Line Type Circuits

Convergence rate is greatly improved by using the new waveform relaxation (WR) methods, known as optimized waveform relaxation (OWR) algorithms, over the classical ones due to the new transmission conditions that exchange more information between subsystems when solving in parallel using multi processors. The new transmission conditions include a free parameter whose value should be chosen in an appropriate way in order to get the best convergence result. In all previous work on OWR for transmission line type circuits, the analysis for finding such value of the free parameter was performed at the continuous level in which any impact of the temporal discretization is neglected. However, it is more realistic to perform the analysis at the discrete level for real computations in a computer when solving time dependant problems. Therefore, choosing transmission line type circuits as our model problem, we perform in this paper the analysis for finding the value of the free parameter appears in the transmission conditions at the discrete level and we study the resulting convergence behavior of the discrete OWR algorithm. We focus on revealing the influence of the time discretization parameter $\triangle t$ on the convergence rate of the OWR algorithm.